Martin's maximum and the non-stationary ideal
英文摘要:
We analyze the non-stationary ideal and the club filter at aleph_1 under MM.
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Shimon Garti, Yair Hayut, Haim Horowitz, Menachem Magidor
We analyze the non-stationary ideal and the club filter at aleph_1 under MM.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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We prove that Martin's Maximum does not imply the Diagonal Reflection Principle for stationary subsets of $[ ω_2 ]^ω$.
Bounded Martin's Maximum with Many Witnesses
我们研究了有界Martin极大值的一个加强,它断言:如果一个关于ω_2 ^V在一个保持扩张的平稳集中成立,那么对于一个序数小于ω_2的平稳集,它在V中成立。我们证明,该原理隐含了全局投影确定性,因此在\mathsf{BMM}的\mathbb{P}_{max}模型中不成立,但该原理对使ω_2 ^V可列共尾的强迫的限制在\mathsf{BMM}模型中确实成立,尽管它不是\mathsf{BMM}的结果。
Martin's maximum revisited
我们提出了几个结果有关的一般理论的固定塔强迫Woodin强迫公理。主要结果是:强迫公理MM^{++}(也称为MM^{+ω_1})决定了H_{ω_2}关于平稳集保持强迫的P_2-理论.我们认为,这是一个接近最佳的推广H_{ω_2}的Woodin的绝对性结果L(R)。
Thin ultrafilters, P-hierarchu and MArtin Axiom
在MA下,我们证明了对于$ω$上的薄集的理想$\cal I$和任意序数$γ\leq ω_1$,存在一个${\cal I}$ -超滤子(在Baumgartner意义下),它属于P-超滤子族的类${\cal P}_γ$.由于这类${\cal P}_2$-超滤子与一类P-点重合,我们的结果推广了Flašková的定理:存在非P-点的${\cal I}$ -超滤子.它也与一个定理有关,该定理指出,在CH下,对于$ω$上的任何高P-理想$\cal I$,都存在一个${\cal I}$ -超滤子,但薄集的理想不是P-理想。
The Stationary Set Splitting Game
\emph{stationary set splitting game}是两个参与者\unspls和\spl之间的长度为$ω_{1}$的完全信息博弈,其中\unspls选择固定的许多可数序数,而\spls试图不断地将它们分成两个固定的部分。我们表明,在ZFC中,有可能迫使任何一个玩家采取获胜策略,或者两者都不采取。这给出了一个新的反例,$Σ^{2}_{2}$极大性与谓词的非平稳理想$ω_{1}$,和一个例子,一个一贯未定的游戏长度$ω_{1}$与支付定义在二阶一元逻辑的顺序。我们还表明,游戏的确定性是符合马丁的公理,但不是马丁的最大值。
Nowhere precipitousness of the non-stationary ideal over P_kappa(lambda)
我们证明了如果lambda是一个强极限奇异基数,kappa是一个正则不可数基数< lambda,那么P_{kappa} lambda上的非平稳理想NS_{kappa lambda}是不陡的.我们还表明,在相同的假设下,P_{kappa} lambda的每一个平稳子集可以划分成lambda^{< kappa}不相交平稳集。
We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $ω_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $MM^{++}$ of Martin's Maximum does not decide whether the restriction of the non-stationary ideal on $ω_2$ to sets of ordinals of countable cofinality is $Δ_1$-definable by formulas with parameters in $H(ω_3)$. The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on $ω_2$ and strong forcing axioms that are compatible with CH. Fin…
The complexity of non-stationary ideals
该研究概述了集合论中不可数正则基数的非平稳理想能否用特定参数的$\Pi_1$公式定义的相关结果,指出该问题与集合论当前多个核心研究主题深度关联。
Martin's Maximum and tower forcing
文献中有几个例子表明基数$κ$的类紧性质导致某些具有临界点$κ$的一般超幂的不良行为(当$κ$是超紧基数时,伯克\cite{MR1472122};当$κ= ω_2$存在强强迫公理时,福尔曼-马吉德\cite{MR1359154})。我们证明了这种现象的更多实例。首先,反射原理(RP)意味着,如果$\vec{\mathcal{I}}$是一座理想之塔,集中在$ω_1$的类$GIC_{ω_1}$上-猜测,内部俱乐部集,那么$\vec{\mathcal{I}}$不是预饱和的(集合是$ω_1$ -猜测,当且仅当其传递崩溃具有$ω_1$ -Hamkins中定义的逼近属性时)\cite{MR2540935})。这个定理,结合\cite{VW_ISP}的工作,表明如果$PFA^+$或$MM$成立,并且有一个不可接近的基数,那么有一个塔的临界点$ω_2$是不预饱和的;此外,这个塔是显着不同的非预饱和塔已知(由福尔曼-马吉德\cite{MR1359154})存在于所有模型的马丁的最大值。强反射原理(SRP)和$ω_2$中的树属性的结合对理想塔有类似的影响,这些理想塔集中在更广泛的$ω_1$类$GIS_{ω_1}$-猜测,内部固定集。最后,我们表明,“presaturated”一词不能被替换为“陡峭的”在上述定理:马丁的最大值(这意味着SRP和树属性在$ω_2$)是一致的陡峭塔$GIC_{ω_1}$。
Some consequences of reflection on the approachability ideal
我们在奇异κ以下正则基数的大基数性质的背景下研究了可逼近理想I[κ^+].作为一个指导性例子,考虑可逼近理想I[ \aleph_{ω+1} ],假设\aleph_ω是强极限。在这种情况下,我们得到俱乐部的许多点在\aleph_{ω+1}的共尾性\aleph_n的一些n>1是可接近的假设联合反射的可数家庭的平稳子集的\aleph_n。这个反射原理在Martin极大值下对所有n>1成立,并且对每个n>1都是等相容的,其中\aleph_n在L中是弱紧的。这表征了马丁极大值模型中的可逼近理想I[ \aleph_{ω+1} ]的结构。
We study several ideal-based constructions in the context of singular stationarity. By combining methods of strong ideals, supercompact embeddings, and Prikry-type posets, we obtain three consistency results concerning mutually stationary sets, and answer a question of Foreman and Magidor concerning stationary sequences on the first uncountable cardinals, $\aleph_n$, $1 \leq n < ω$.
For $κ$ regular and uncountable we define variants of the classical cardinal characteristics modulo the non-stationary ideal.
Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal $κ$. We show the consistency of $E^{λ^{++},λ^{++}}_{λ\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{λ^{++}}_λ$ in the space $(λ^{++})^{λ^{++}}$, being continuously reducible to $E^{2,λ^{++}}_{λ^+\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{λ^{++}}_{λ^+}$ in the space $2^{λ^{++}}$. Then we show the consistency of $E^{2,κ}_{reg}$…
We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the $ω_1$-approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal $I[ω_2]$ does not have a maximal set modulo clubs.
Uniformization of ladder system colorings and stationary precaliber forcings
本文研究了梯子系统染色均匀化变体与 Martin 公理片段的关系,提出了平稳预 caliber $\aleph_1$ 性质的力迫公理,并证明了其与 $σ$-链接偏序集力迫公理的分离。
We show that in extender models there are no generic embeddings with critical point $ω_1$ that resemble the stationary tower at the second Woodin cardinal.
Club-guessing, stationary reflection, and coloring theorems
我们得到强染色定理在继承人的奇异基数从失败的某些情况下,同时反映的平稳集。一路上,我们在俱乐部猜测和理想的一般理论中建立了新的结果。
We show that, under the assumption of the existence of $M_1^{\#}$, there exists a model on which the restricted nonstationary ideal $\hbox{NS} \upharpoonright A$ is $\aleph_2$-saturated, for $A$ a stationary co-stationary subset of $ω_1$, while the full nonstationary ideal $\hbox{NS}$ can be made $Δ_1$ definable with $K_{ω_1}$ as a parameter. Further we show, again under the assumption of the existence of $M_1^{\#}$ that there is a model of set theory such that $\hbox{NS}$ is $\aleph_2$-saturated and such that there is lightface $Σ^1_4$-definable well-order on the reals. This result is optimal…
Echeloned saturation and forcing axioms
针对Paul Larson的问题,研究结合不同力迫公理与集合论假设,证明特定条件下ℵ₁上无弱Laver理想,以及Baumgartner公理下ℵ₂上无Laver理想。
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